Abstract

Low-density parity-check (LDPC)-coded orthogonal frequency division multiplexing (OFDM) is studied as an efficient coded modulation scheme suitable for simultaneous chromatic dispersion and polarization mode dispersion (PMD) compensation. We show that, for aggregate rate of 10 Gb/s, accumulated dispersion over 6500 km of SMF and differential group delay of 100 ps can be simultaneously compensated with penalty within 1.5 dB (with respect to the back-to-back configuration) when training sequence based channel estimation and girth-10 LDPC codes of rate 0.8 are employed.

© 2008 Optical Society of America

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References

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  1. R. Prasad, OFDM for Wireless Communications Systems (Artech House, Boston 2004).
  2. I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
    [CrossRef] [PubMed]
  3. W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
    [CrossRef]
  4. A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.
  5. I. B. Djordjevic and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
    [CrossRef]
  6. I. B. Djordjevic, "PMD compensation in fiber-optic communication systems with direct detection using LDPC-coded OFDM," Opt. Express 15, 3692-3701 (2007).
    [CrossRef] [PubMed]
  7. W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photon. Technol. Lett. 19, 134-136 (2006).
    [CrossRef]
  8. S. L. Jansen, I. Morita, N. Takeda, H. Tanaka, "20-Gb/s OFDM transmission over 4,160-km SSMF enabled by RF-pilot tone phase compensation," in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP15, March 25-29, 2007, Anaheim, CA, USA.
  9. B. J. Schmidt, A. J. Lawery, J. Amstrong, "Experimental demonstration of 20 Gbit/s direct-detection optical OFDM and 12 Gbit/s with a colorless transmitter," in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP18, March 25-29, 2007, Anaheim, CA, USA.
  10. A. Lowery, "Nonlinearity and its compensation in optical OFDM systems," presented at ECOC 2007 Worksop 5 (Electronic signal processing for transmission impairment mitigation: future challenges).
  11. I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
    [CrossRef]
  12. W. Shieh, X. Yi, Y. Ma, and Y. Tang, "Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems," Opt. Express 15, 9936-9947 (2007).
    [CrossRef] [PubMed]
  13. I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
    [CrossRef]
  14. M. P. C. Fossorier, "Quasi-cyclic low-density parity-check codes from circulant permutation matricies," IEEE Trans. Inform. Theory 50, 1788-1794 (2004).
    [CrossRef]
  15. O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
    [CrossRef]
  16. J. L. Fan, "Array codes as low-density parity-check codes," in Proc. 2nd Int. Symp. Turbo Codes and Related Topics, Brest, France, pp. 543-546, Sept. 2000.
  17. D. J. C. MacKay, "Good error correcting codes based on very sparse matrices," IEEE Trans. Inform. Theory 45, 399-431 (1999).
    [CrossRef]
  18. R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory IT-27, 533-547 (1981).
    [CrossRef]
  19. T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
    [CrossRef]
  20. N. Cvijetic, L. Xu, and T. Wang, "Adaptive PMD Compensation using OFDM in Long-Haul 10Gb/s DWDM Systems," in Optical Fiber Comm. Conf., Paper OTuA5, Anaheim, CA (2007).

2007 (3)

2006 (5)

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
[CrossRef]

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

I. B. Djordjevic and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photon. Technol. Lett. 19, 134-136 (2006).
[CrossRef]

I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
[CrossRef] [PubMed]

2004 (3)

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

M. P. C. Fossorier, "Quasi-cyclic low-density parity-check codes from circulant permutation matricies," IEEE Trans. Inform. Theory 50, 1788-1794 (2004).
[CrossRef]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
[CrossRef]

1999 (1)

D. J. C. MacKay, "Good error correcting codes based on very sparse matrices," IEEE Trans. Inform. Theory 45, 399-431 (1999).
[CrossRef]

1981 (1)

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory IT-27, 533-547 (1981).
[CrossRef]

Akita, M.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Athaudage, C.

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

Batshon, H. G.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
[CrossRef]

Chilappagari, S. K.

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
[CrossRef]

Cvijetic, M.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
[CrossRef]

Djordjevic, I. B.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
[CrossRef]

I. B. Djordjevic, "PMD compensation in fiber-optic communication systems with direct detection using LDPC-coded OFDM," Opt. Express 15, 3692-3701 (2007).
[CrossRef] [PubMed]

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
[CrossRef]

I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
[CrossRef] [PubMed]

I. B. Djordjevic and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
[CrossRef]

Fossorier, M. P. C.

M. P. C. Fossorier, "Quasi-cyclic low-density parity-check codes from circulant permutation matricies," IEEE Trans. Inform. Theory 50, 1788-1794 (2004).
[CrossRef]

Fujita, H.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Kobayashi, T.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Kubo, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Kuno, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Ma, Y.

MacKay, D. J. C.

D. J. C. MacKay, "Good error correcting codes based on very sparse matrices," IEEE Trans. Inform. Theory 45, 399-431 (1999).
[CrossRef]

Milenkovic, O.

O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
[CrossRef]

Miyata, Y.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Mizuochi, T.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Motoshima, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Ouchi, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Sankaranarayanan, S.

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
[CrossRef]

Shieh, W.

W. Shieh, X. Yi, Y. Ma, and Y. Tang, "Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems," Opt. Express 15, 9936-9947 (2007).
[CrossRef] [PubMed]

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photon. Technol. Lett. 19, 134-136 (2006).
[CrossRef]

Shimizu, K.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Tagami, H.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Tang, Y.

Tanner, R. M.

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory IT-27, 533-547 (1981).
[CrossRef]

Vasic, B.

I. B. Djordjevic and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

I. B. Djordjevic and B. Vasic, "Orthogonal frequency-division multiplexing for high-speed optical transmission," Opt. Express 14, 3767-3775 (2006).
[CrossRef] [PubMed]

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
[CrossRef]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
[CrossRef]

Wang, T.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
[CrossRef]

Xu, L.

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
[CrossRef]

Yi, X.

Yoshida, H.

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

Electron. Lett. (1)

W. Shieh and C. Athaudage, "Coherent optical frequency division multiplexing," Electron. Lett. 42, 587-589 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, "Low-density parity-check codes for 40 Gb/s optical transmission systems," IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).
[CrossRef]

O. Milenkovic, I. B. Djordjevic, and B. Vasic, "Block-circulant low-density parity-check codes for optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).
[CrossRef]

T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida, H. Fujita, M. Akita, and K. Motoshima, "Forward error correction based on block turbo code with 3-bit soft decision for 10-Gb/s optical communication systems," IEEE J. Sel. Top. Quantum Electron. 10, 376-386 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

I. B. Djordjevic and B. Vasic, "100 Gb/s transmission using orthogonal frequency-division multiplexing," IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).
[CrossRef]

I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, "PMD compensation by LDPC-coded turbo equalization," IEEE Photon. Technol. Lett. 19, 1163 - 1165 (2007).
[CrossRef]

W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photon. Technol. Lett. 19, 134-136 (2006).
[CrossRef]

IEEE Trans. Inf. Theory (1)

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory IT-27, 533-547 (1981).
[CrossRef]

IEEE Trans. Inform. Theory (2)

D. J. C. MacKay, "Good error correcting codes based on very sparse matrices," IEEE Trans. Inform. Theory 45, 399-431 (1999).
[CrossRef]

M. P. C. Fossorier, "Quasi-cyclic low-density parity-check codes from circulant permutation matricies," IEEE Trans. Inform. Theory 50, 1788-1794 (2004).
[CrossRef]

Opt. Express (3)

Other (7)

R. Prasad, OFDM for Wireless Communications Systems (Artech House, Boston 2004).

A. J. Lowery, L. Du, and J. Armstrong, "Orthogonal frequency division multiplexing for adaptive dispersion compensation in long haul WDM systems," in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.

S. L. Jansen, I. Morita, N. Takeda, H. Tanaka, "20-Gb/s OFDM transmission over 4,160-km SSMF enabled by RF-pilot tone phase compensation," in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP15, March 25-29, 2007, Anaheim, CA, USA.

B. J. Schmidt, A. J. Lawery, J. Amstrong, "Experimental demonstration of 20 Gbit/s direct-detection optical OFDM and 12 Gbit/s with a colorless transmitter," in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP18, March 25-29, 2007, Anaheim, CA, USA.

A. Lowery, "Nonlinearity and its compensation in optical OFDM systems," presented at ECOC 2007 Worksop 5 (Electronic signal processing for transmission impairment mitigation: future challenges).

J. L. Fan, "Array codes as low-density parity-check codes," in Proc. 2nd Int. Symp. Turbo Codes and Related Topics, Brest, France, pp. 543-546, Sept. 2000.

N. Cvijetic, L. Xu, and T. Wang, "Adaptive PMD Compensation using OFDM in Long-Haul 10Gb/s DWDM Systems," in Optical Fiber Comm. Conf., Paper OTuA5, Anaheim, CA (2007).

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Figures (6)

Fig. 1.
Fig. 1.

LDPC-coded OFDM: (a) transmitter configuration, and (b) receiver configuration. LDPCE-LDPC encoder, LDPCD-LDPC decoder, S/P-serial-to-parallel converter, MZM-Mach-Zehnder modulator, SMF-single-mode optical fiber, PD-photodetector, DSB-double-sideband, SSB-single-sideband.

Fig. 2.
Fig. 2.

Waveforms and PSDs of SSB QPSK-OFDM signal at different points during transmission for electrical SNR (per bit) of 6dB. (f c denotes the optical carrier frequency, LD denotes the laser diode).

Fig. 3.
Fig. 3.

The girth-10 LDPC code against RS, concatenated RS, turbo-product, and girth-8 LDPC codes on an AWGN channel model.

Fig. 4.
Fig. 4.

BER performance of LDPC-coded OFDM system with aggregate rate of 10 Gb/s, for DGD of 100 ps.

Fig. 5.
Fig. 5.

BER performance of LDPC-coded OFDM system with aggregate rate of 10 Gb/s, after 6500 km of SMF.

Fig. 6.
Fig. 6.

BER performance of LDPC-coded OFDM system with aggregate rate of 10 Gb/s, after 6500 km of SMF and for DGD of 100 ps.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

s ( t ) = s OFDM ( t ) + D ,
i ( t ) = R PIN { [ s OFDM ( t ) + D ] * h ( t ) + N ( t ) } 2 ,
r ( t ) = [ i ( t ) k RF cos ( ω RF t ) ] * h e ( t ) + n ( t ) ,
d { 1 + r r 2 ( ( r 1 ) ( g 2 ) 4 1 ) , g 2 = 2 m + 1 1 + r r 2 ( ( r 1 ) ( g 2 ) 4 1 ) + ( r 1 ) ( g 2 ) 4 , g 2 = 2 m
H = [ I P c 0 P 2 c 0 P ( q 1 ) c 0 I P c 1 P 2 c 1 P ( q 1 ) c 1 I P c r 1 P 2 c r 1 P ( q 1 ) c r 1 ] ,
c i 1 j 1 + c i 2 j 2 + + c i k j k = c i 1 j 2 + c i 2 j 3 + + c i k j 1 mod q ,
R n q r q n q = 1 r n ,
E ( t ) = FT 1 { FT { E 0 [ 1 κ k e j δ ] [ s OFDM ( t ) + b ] } e j ( β 2 ω 2 2 β 3 ω 3 6 ) L tot } + [ N x ( t ) N y ( t ) ] ,
β 2 L tot Δ ω + DGD max = c f 2 D t N FFT Δ f + DGD max T G ,
Y i , k = h i e j θ i e j ϕ k X i , k + n i , k ,
X ̂ i , k = ( h i * h i 2 ) e j θ i e j ϕ k Y i , k .
λ ( q ) = ( Re [ X ̂ i , k ] Re [ QAM ( map ( q ) ) ] ) 2 N 0 ( Im [ X ̂ i , k ] Im [ QAM ( map ( q ) ) ] ) 2 N 0 ; q = 0 , 1 , , 2 b 1
L ( v ̂ j ) = log q : v j = 0 exp [ λ ( q ) ] q : v j = 1 exp [ λ ( q ) ] ,

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